# Removing Even Crossings

1 Illinois Institute of Technology - Department of Applied Mathematics 2 DePaul University - Department of Computer Science 3 Rochester - Computer Science Department

Abstract : An edge in a drawing of a graph is called $\textit{even}$ if it intersects every other edge of the graph an even number of times. Pach and Tóth proved that a graph can always be redrawn such that its even edges are not involved in any intersections. We give a new, and significantly simpler, proof of a slightly stronger statement. We show two applications of this strengthened result: an easy proof of a theorem of Hanani and Tutte not using Kuratowski-s theorem, and the result that the odd crossing number of a graph equals the crossing number of the graph for values of at most $3$. We begin with a disarmingly simple proof of a weak but standard version of the theorem by Hanani and Tutte.

Keywords : Hanani-s theorem Tutte-s theorem even crossings crossing number odd crossing number independent odd crossing number

Author: Michael J. Pelsmajer - Marcus Schaefer - Daniel Štefankovič -

Source: https://hal.archives-ouvertes.fr/